Matrix Computations (Johns Hopkins Studies in the Mathematical Sciences, 3)
The fourth edition of Gene H. Golub and Charles F. Van Loan's classic is an essential reference for computational scientists and engineers in addition to researchers in the numerical linear algebra community. Anyone whose work requires the solution to a matrix problem and an appreciation of its mathematical properties will find this book to be an indispensible tool.
This revision is a cover-to-cover expansion and renovation of the third edition. It now includes an introduction to tensor computations and brand new sections on • fast transforms• parallel LU• discrete Poisson solvers• pseudospectra• structured linear equation problems• structured eigenvalue problems• large-scale SVD methods• polynomial eigenvalue problems
Matrix Computations is packed with challenging problems, insightful derivations, and pointers to the literature―everything needed to become a matrix-savvy developer of numerical methods and software.
Why Read This Book
You will read this book to gain a deep, implementation-oriented understanding of the matrix algorithms that underlie modern signal processing, communications, and scientific computing. It pairs rigorous analysis with practical algorithmic detail — so you can both reason about numerical stability and deploy robust, high-performance routines for FFTs, SVD, eigenproblems and large-scale linear systems.
Who Will Benefit
Advanced engineers, researchers, and graduate students in DSP, communications, radar, and computational science who need a rigorous reference for matrix algorithms and their numerical properties.
Level: Advanced — Prerequisites: Undergraduate linear algebra (matrix factorizations, eigenvalues), basic numerical analysis, and familiarity with algorithmic programming (MATLAB/NumPy/Fortran/C) for implementing and testing algorithms.
Key Takeaways
- Understand core matrix factorizations (LU, QR, Cholesky) and when to choose each for stability and performance
- Analyze and solve eigenvalue and singular value problems relevant to spectral analysis, PCA, and subspace methods
- Implement robust algorithms for large-scale linear systems and least-squares problems used in adaptive filtering and parameter estimation
- Apply fast transforms and structured-matrix techniques (e.g., FFT-related methods, Toeplitz/Poisson solvers) to accelerate DSP and radar computations
- Evaluate numerical stability, conditioning, and pseudospectra to diagnose and mitigate errors in signal-processing pipelines
Topics Covered
- Preliminaries: Matrix Algebra, Norms, and Conditioning
- Matrix Factorizations: LU, Cholesky, and QR
- Orthogonality and Least Squares Methods
- Eigenvalue Problems: Symmetric and Nonsymmetric Algorithms
- Singular Value Decomposition and Applications
- Iterative Methods for Linear Systems and Eigenproblems
- Structured Matrices, Fast Transforms, and Toeplitz/PDE Solvers
- Large-Scale SVD and Eigenvalue Methods; Polynomial Eigenvalue Problems
- Parallel Algorithms and Parallel LU
- Pseudospectra and Stability Analysis
- Tensor Computations and Higher-Order Generalizations
- Appendices: Implementation Notes, LAPACK/BLAS References, and Bibliography
Languages, Platforms & Tools
How It Compares
More comprehensive and implementation-focused than Trefethen & Bau's Numerical Linear Algebra; complements Numerical Recipes by emphasizing rigorous algorithmic analysis and large-scale, structured problems.
